The given displacement at time, t, is
p(t) = 2 sin(t³) + 5 cos(t³)

The initial equilibrium position is
p(0) = 5

To determine future equilibrium postions, define
f(t) = p(t) - 5 = 2 sin(t³) + 5 cos(t³ - 5
The derivative of f(t) is
f'(t) = (3t²)[2 cos(t³) - 5sin(t³)]

Equilibrium is established when f(t) = 0.
To solve this equation numerically, we shall use the Newton-Raphson method, given by.
t(n+1) = t(n) - f[t(n)]/f'[(t(n)], n=0,1,2, ...,

As a guess, let (0) = 1.
The iterative solution for t is shown below.

n t(n)
--- -----------
0 1.0000
1 0.9344
2 0.9147
3 0.9130
4 0.9130

The solution converges rapidly to t = 0.913 s.
The graphical solution (shown below) confirms the numerical solution.

Answer:
The weight first reaches the equilibrium position in 0.913 sec.